What is the statement of the Stokes theorem
In this article, the Stokes' theorem treated. First the general Stokes' theorem formulated before briefly on its special cases Main theorem of differential and integral calculus (HDI) as well as the Gaussian integral theorem is received. In addition, the classic integral theorem from Stokes as a further special case of the general will be examined in more detail. Finally, the calculation takes place two examples.
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- Stokes general integral theoremin the text
- Underlying topological principlein the text
- Stokes' theorem as a classical Stokes integral theoremin the text
- in the text
Stokes general integral theorem
If from Stokes' theorem is in question, in most cases it is the classic Stokes integral theorem meant. It represents a special case of the general integral theorem of Stokes which reads as follows:
Be open and a oriented -dimensional submanifold With as a continuously differentiable -Shape in . Then applies to everyone compact amount with a smooth edge
in which the induced orientation carries and the outer derivative of designated.
Underlying topological principle
The Stokes' theorem lies that topological principle is based on the fact that when paving a piece of land by identically oriented "paving stones" the inner ways in opposite direction will go through, which leads to their Mutually cancel contributions to the line integral and only that Contribution of the boundary curve remains.
Main theorem of differential and integral calculus as a special case
For degenerate the general integral theorem of Stokes to the Law of differential and integral calculus: Be an open interval and a continuously differentiable function. Then the following applies:
Gauss integral theorem as a special case
Another special case follows from the general integral theorem of Stokes the Gaussian integral theorem. To show that will be chosen and be it
i.e. with the continuously differentiable vector field . The roof shows over indicates that this factor must be omitted. Be also the outer unit normal field, then applies
With also results
Ultimately, this gives that Gaussian integral theorem
Stokes' theorem as a classical Stokes integral theorem
Often, and especially in technical courses and physics, there is talk of Stokes' theorem. This is usually the classic integral theorem from Stokes meant which one too Kelvin-Stokes theorem or Rotation set is called. Together with the Gaussian integral theorem he plays an essential role in the formulation of the Maxwell's equations in the integral form.
Special case of Stokes' general integral theorem
The classic theorem of Stokes results like that HDI and the Gaussian integral theorem as Special case of Stokes' general integral theorem. In this case the open amount as well as that continuously differentiable vector field considered. put one two-dimensional submanifold whose orientation through the Unit normal field is given. On the submanifold be on Compact form given which one smooth edge own. This in turn is through the Unit tangent field oriented. With the in continuously differentiable Pfaffian form
thus results in the Stokes' theorem:
In another notation it reads:
The following can be seen:
The Stokes' theorem states that a Area integral about the rotation one Vector field under certain conditions in a closed curve integral over to the curve tangential component of Vector field can be converted. The curve traversed must be the Edge of the area under consideration correspond.
In the following, the Stokes' theorem be proven. For this proof, however, there is a small one condition to the surface posed. This should be the graph a function be which over one area in the Level is defined. With and be the Projections of and the counterclockwise edge on the -Level. be through
parameterized, from which with the help of the Chain rule follows:
For the im Stokes' theorem looked at Curve integral:
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